Polynomial functions on Young diagrams arising from bipartite graphs

264 Maciej Dołęga and Piotr Śniady
edges are replaced by single edges). The edge resulting from gluing f and z will be decorated by z. More
generally, if G is a linear combination of bipartite graphs, this definition extends by linearity.
We also define
∂ y G = ∑ G f≡z
f
which is a formal sum which runs over all edges f ≠ z which share a common white vertex with the edge
z.
Conjecture 4.1 Let G be a linear combination of bipartite graphs with a property that
(∂ x + ∂ y ) ∂ z G = 0.
Then for any integer k ≥ 1 (
∂
k
x − (−∂ y ) k) ∂ z G = 0.
We are able to prove Conjecture 4.1 under some additional assumptions, however we believe it is true in
general.
5 Characterization of <strong>functions</strong> arising from bipartite graphs which
are polynomial
5.1 The main result
Theorem 5.1 Let G be a linear combination of bipartite graphs such that
(
∂
k
x − (−∂ y ) k) ∂ z G = 0 (3)
for any integer k > 0. Then λ ↦→ NG λ is a polynomial function on the set of Young diagrams.
The main idea of the proof is to find a connection between content-derivative of a function N G and a
combinatorial derivation of the underlying linear combination of bipartite graphs G; we present it in the
following.
5.2 Colorings of bipartite graphs with decorated edges
Let a Young diagram λ and a bipartite graph G be given. If an edge of G is decorated by a real number z,
we decorate its white end by the number ω(z)+z
2
(which is the x-coordinate of the point at the profile of λ
with contents equal to z) and we decorate its black end by the number ω(z)−z
2
(which is the y-coordinate
of the point at the profile of λ with contents equal to z). If some disjoint edges are decorated by n real
numbers z 1 , . . . , z n , then we decorate white and black vertices in an analogous way.
For a bipartite graph G with some disjoint edges decorated we define N G (λ), the number of colorings
of λ, as the volume of the set of <strong>functions</strong> from undecorated vertices to R + such that these <strong>functions</strong>
extended by values of decorated vertices are compatible with λ.
We will use the following lemma:
Lemma 5.2 Let (G, z) be a bipartite graph with one edge decorated by a real number z. Then
• d dz N (G,z)(λ) = ω′ (z)+1
2
N ∂x(G,z)(λ) + ω′ (z)−1
2
N ∂y(G,z)(λ),